# zbMATH — the first resource for mathematics

The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I: Compactness methods. (English) Zbl 0889.35045
It is studied the Cauchy problem for the equation $\partial_t u = \gamma u +(a+ i \alpha) \Delta u - (b + i \beta u g(|u|^2),$ $$a>0$$, $$b>0$$, $$g \geq 0$$, in arbitrary spatial dimension. The initial data and the solutions under consideration belong to local spaces, without any restriction at infinity. It is proved the existence of solution globally defined in time with local regularity corresponding to $$L^r$$, $$r \geq 2$$, and $$H^1$$. Some uniqueness results are presented as well.

##### MSC:
 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
arbitrary spatial dimension
Full Text:
##### References:
 [1] Brézis, H, Analyse fonctionnelle, (1983), Masson Paris · Zbl 0511.46001 [2] Cazenave, T, An introduction to nonlinear Schrödinger equations, () · Zbl 0528.35008 [3] Collet, P, Thermodynamic limit of the Ginzburg-Landau equation, Nonlinearity, 7, 1175-1190, (1994) · Zbl 0803.35066 [4] Cross, M.C; Hohenberg, P.C, Pattern formation outside of equilibrium, Rev. modern phys., 65, 851-1089, (1993) · Zbl 1371.37001 [5] Doering, C.R; Gibbon, J.D; Levermore, C.D, Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71, 285-318, (1994) · Zbl 0810.35119 [6] Ghidaglia, J.M; Héron, B, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica D, 28, 282-304, (1987) · Zbl 0623.58049 [7] Giga, Y, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. differential equations, 61, 186-212, (1986) · Zbl 0577.35058 [8] Ginibre, J; Velo, G, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. inst. H. Poincaré anal. non linéaire, 2, 309-327, (1985) · Zbl 0586.35042 [9] Ginibre, J; Velo, G, The Cauchy problem in local spacesfor the complex Ginzburg-Landau equation. part II. contraction methods, (1996), preprint · Zbl 0889.35045 [10] Kato, T, Nonlinear Schrödinger equations, (), 218-263 [11] Kozono, K; Yamazaki, M, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. partial differential equation, 19, 959-1014, (1994) · Zbl 0803.35068 [12] Levermore, C.D; Oliver, M, The complex Ginzburg-Landau equation as a model problem, (), 141-189 · Zbl 0845.35003 [13] Lions, J.L, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, (1969), Dunod Paris · Zbl 0189.40603 [14] Lions, J.L; Magenes, E, () [15] Segal, I.E, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. soc. math. France, 91, 129-135, (1963) · Zbl 0178.45403 [16] Strauss, W, On weak solutions of semilinear hyperbolic equations, An. acad. brasil. cienc., 42, 645-651, (1970) · Zbl 0217.13104 [17] Tartar, L, Topics in nonlinear analysis, Publ. mat. orsay, (1978) · Zbl 0401.35014 [18] Temam, R, Navier-Stokes equations, (1984), North-Holland Amsterdam · Zbl 0572.35083 [19] von Wahl, W, The equations of Navier-Stokes and abstract parabolic equations, (1985), Vieweg Braunschweig · Zbl 1409.35168 [20] Weissler, F.B, Local existence and non existence for semilinear parabolic equations in Lp, Indiana univ. math. J., 29, 79-102, (1980) · Zbl 0443.35034 [21] Weissler, F.B, Existence and non existence of global solutions for a semilinear heat equation, Israel J. math., 38, 29-40, (1981) · Zbl 0476.35043 [22] Yosida, K, Functional analysis, (1978), Springer Berlin · Zbl 0152.32102 [23] Mielke, A; Schneider, G, D attractors for modulation equations on unbounded domains, exisence and comparison, Nonlinearity, 8, 743-768, (1995) · Zbl 0833.35016 [24] Snoussi, S, Étude du comportement asymptotique des solutions d’une équation de Ginzburg-Landau généralisée, Thesis, (1996), Orsay
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.